1. Entanglement and Optimal Strings of Qubits for Memory Channels Laleh Memarzadeh Sharif University of Technology IICQI 7-10 Sept 2007 Kish Island

2. Outline  Classical Capacity of Quantum Channels  The basic question: Does entanglement enhances classical capacity?  Definition of Memory Channels  Previous results  Our question  Our Result

3. Definition of a channel  Completely Positive  Trace preserving Quantum channel:

4. Channel Capacity Input state

5. Optimal Input Ensemble of States A B C DA B C D Separable States Maximally Entangled States What is the Optimal ensemble of input states?

6. Product Channels  Uncorrelated channels: No advantage in using entangled states

7. Memory Channels  Memory Channel Uncorrelated noise Full Memory

8. Previous Results  Depolarizing channel (D.Brub, L.Faoro, C. Macchiavello, G. Palma 2002)  Symmetric Pauli channel (C. Macchiavello, GPalma, S. Virmani,2004).  Guassian channels (N.Cerf, J. Clavareau, C. Macchiavello. J. Roland,2005).  ……… Separable states are optimal input states Entangled states are optimal input states

9. What Is Our Question?  Does encoding information in arbitrary long entangled state enhance the mutual information?

10. The significance of this question  Classical Capacity of the Channel: Input Length  Optimize the mutual information over all ensembles of n qubit states.

11. Gaining an insight into this problem For the Pauli channels Optimization of mutual information Is equivalent to Finding a single pure state which minimize the output entropy  Kraus operators of the channel commute or anti-commute  They form an irreps of the Pauli group  Convexity property of entropy

12. Typical long strings  Separable states  GHZ states

13. Output Entropy Strings of odd length No advantage in using entangled input states

14. Output Entropy Strings of even length Encoding data in entangled input states is useful for

15. Critical Memory vs string length When n increases V. Karimipour, L. Memarzadeh, Phys. Rev. A (2006)

16. Final words:  Even for memory channels we can’t be sure that there is any advantage in using entangled states for encoding information.  Do you need a flight from Kish to Dubai?  If yes, please send me your exact flight information from Open problems:

17. Thanks for Your Attention